# Area of the Largest Triangle inscribed in a Hexagon

Given here is a regular hexagon, of side length a, the task is to find the area of the biggest triangle that can be inscribed within it.

Examples:

Input:  a = 6
Output: area = 46.7654

Input: a = 8
Output: area = 83.1384

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach:

It is very clear that the biggest triangle that can be inscribed within the hexagon is an equilateral triangle.
In triangle ACD,
following pythagorus theorem,
(a/2)^2 + (b/2)^2 = a^2
b^2/4 = 3a^2/4
So, b = a√3

Therefore, area of the triangle, A = √3(a√3)^2/4= 3√3a^2/4

Below is the implementation of the above approach:

## C++

 // C++ Program to find the biggest triangle // which can be inscribed within the hexagon #include using namespace std;    // Function to find the area // of the triangle float trianglearea(float a) {        // side cannot be negative     if (a < 0)         return -1;        // area of the triangle     float area = (3 * sqrt(3) * pow(a, 2)) / 4;        return area; }    // Driver code int main() {     float a = 6;     cout << trianglearea(a) << endl;        return 0; }

## Java

 // Java Program to find the biggest triangle // which can be inscribed within the hexagon    import java.io.*;    class GFG {        // Function to find the area // of the triangle static double trianglearea(double a) {        // side cannot be negative     if (a < 0)         return -1;        // area of the triangle     double area = (3 * Math.sqrt(3) * Math.pow(a, 2)) / 4;        return area; }        public static void main (String[] args) {         double a = 6;         System.out.println (trianglearea(a));        } //This Code is contributed by Sachin..        }

## Python3

 # Python3 Program to find the biggest triangle # which can be inscribed within the hexagon import math    # Function to find the area # of the triangle def trianglearea(a):        # side cannot be negative     if (a < 0):         return -1;        # area of the triangle     area = (3 * math.sqrt(3) * math.pow(a, 2)) / 4;        return area;    # Driver code a = 6; print(trianglearea(a))    # This code is contributed  # by Akanksha Rai

## C#

 // C# Program to find the biggest triangle  // which can be inscribed within the hexagon    using System;    class GFG {         // Function to find the area  // of the triangle  static double trianglearea(double a)  {         // side cannot be negative      if (a < 0)          return -1;         // area of the triangle      double area = (3 * Math.Sqrt(3) * Math.Pow(a, 2)) / 4;         return Math.Round(area,4);  }         public static void Main () {          double a = 6;          Console.WriteLine(trianglearea(a));         }          // This code is contributed by Ryuga    }

## PHP



Output:

46.7654

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